Summary of the Kolmogorov-Obukhov (1941) theory: overview. In the last three posts we have summarised various aspects of the Kolmogorov-Obukhov (1941) theory. When considering this theory, the following things need to be borne in mind. [a] Whether we are working in -space or -space matters. See my posts of 8 April and 15 April 2021Continue reading Summary of the Kolmogorov-Obukhov (1941) theory: overview.

Summary of the Kolmogorov-Obukhov (1941) theory. Part 3: Obukhov’s theory in k-space. Obukhov is regarded as having begun the treatment of the problem in wavenumber space. In [1] he referred to an earlier paper by Kolmogorov for the spectral decomposition of the velocity field in one dimension and pointed out that the three-dimensional case isContinue reading Summary of the Kolmogorov-Obukhov (1941) theory. Part 3: Obukhov’s theory in k-space.

Summary of the Kolmogorov-Obukhov (1941) theory. Part 2: Kolmogorov’s theory in x-space. Kolmogorov worked in -space and his two relevant papers are cited below as [1] (often referred to as K41A) and [2] (K41B). We may make a pointwise summary of this work, along with more recent developments as follows. [a] In K41A, Kolmogorov introducedContinue reading Summary of the Kolmogorov-Obukhov (1941) theory. Part 2: Kolmogorov’s theory in x-space.

Summary of Kolmogorov-Obukhov (1941) theory. Part 1: some preliminaries in -space and -space. Discussions of the Kolmogorov-Obukhov theory often touch on the question: can the two-thirds law; or, alternatively, the minus five-thirds law, be derived from the equations of motion (NSE)? And the answer is almost always: ‘no, they can’t’! Yet virtually every aspect ofContinue reading Summary of Kolmogorov-Obukhov (1941) theory. Part 1: some preliminaries in x-space and k-space.

The importance of terminology: stationarity or equilibrium? When I began my post-graduate research in 1966, I found that I immediately had to get used to a new terminology. For instance, concepts like homogeneity and isotropy were a definite novelty. In physics one takes these for granted and they are never mentioned. Indeed the opposite isContinue reading The importance of terminology: stationarity or equilibrium?

Turbulence in a box. When the turbulence theories of Kraichnan, Edwards, Herring, and so on, began attracting attention in the 1960s, they also attracted attention to the underlying ideas of homogeneity, isotropy, and Fourier analysis of the equations of motion. These must have seemed very exotic notions to the fluid dynamicists and engineers who workedContinue reading Turbulence in a box.

Large-scale resolution and finite-size effects. This post arises out of the one on local isotropy posted on 21 October 2021; and in particular relates to the comment posted by Alex Liberzon on the need to choose the size of volume within which Kolmogorov’s assumptions of localness may hold. In fact, as is so often theContinue reading Large-scale resolution and finite-size effects.

The second-order structure function corrected for systematic error. In last week’s post, we discussed the corrections to the third-order structure function arising from forcing and viscous effects, as established by McComb et al [1]. This week we return to that reference in order to consider the effect of systematic error on the second-order structure function,Continue reading The second-order structure function corrected for systematic error.

Viscous and forcing corrections to Kolmogorov’s ‘4/5’ law. The Kolmogorov `4/5′ law for the third-order structure function is widely regarded as the one exact result in turbulence theory. And so it should be: it has a straightforward derivation from the Karman-Howarth equation (KHE), which is an exact energy balance derived from the Navier-Stokes equation. Nevertheless,Continue reading Viscous and forcing corrections to Kolmogorov’s ‘4/5’ law.

Local isotropy, local homogeneity and local stationarity. In last week’s post I reiterated the argument that the existence of isotropy implies homogeneity. However, Alex Liberzon commented that there could be inhomogeneous flows that exhibited isotropy on scales that were small compared to the overall size of the flow. This comment has the great merit ofContinue reading Local isotropy, local homogeneity and local stationarity.

Is isotropy the same as spherical symmetry? To which you might be tempted to reply: ‘Who ever thought it was?’ Well, I don’t know for sure, but I’ve developed a suspicion that such a misconception may underpin the belief that it is necessary to specify that turbulence is homogeneous as well as isotropic. When IContinue reading Is isotropy the same as spherical symmetry?

Various kinds of turbulent dissipation? The current interest in Onsager’s conjecture (see my blog of 23 September 2021) has sparked my interest in the nature of turbulent dissipation. Essentially a fluid only moves because a force acts on it and does work to maintain it in motion. The effect of viscosity is to convert thisContinue reading Various kinds of turbulent dissipation?

Superstitions in turbulence theory 2: that intermittency destroys scale-invariance! At the moment I am busy revising a paper (see [1] below) in order to meet the comments of the referees. As is so often the case, Referee 1 is supportive and Referee 2 is hostile. Naturally, Referee 2 writes at great length, so it isContinue reading Superstitions in turbulence theory 2: that intermittency destroys scale-invariance!

Superstitions in turbulence theory 1: the infinite Re limit of the Navier-Stokes equation is the Euler equation! I recently posted blogs about the Onsager conjecture [1]; the need to take limits properly (Onsager didn’t!); and the programme at MSRI Berkeley, which referred to the Euler equation as the infinite Reynolds number limit, in a seriesContinue reading Superstitions in turbulence theory 1: the infinite Re limit of the Navier-Stokes equation is the Euler equation!

Peer review: the role of the referee. In earlier years I used to get the occasional phone call from George Batchelor, at that time the editor of Journal of Fluid Mechanics, asking for suggestions of new referees on the statistical theory of turbulence. To avoid confusion I should point out that by this I meanContinue reading Peer review: the role of the referee.

Peer review: The role of the author. I have previously posted on the role of the editor (see my blog on 09/07/2020) and had intended to go on to discuss the role of the referee. However, before doing that it occurred to me that it might be helpful to first discuss the role of theContinue reading Peer review: The role of the author.

The exactness of mathematics and the inexactness of physics. This post was prompted by something that came up in a previous one (i.e. see my blog on 12 August 2021), where I commented on the fact that an anonymous referee did not know what to make of an asymptotic curve. The obvious conclusion from thisContinue reading The exactness of mathematics and the inexactness of physics.

Nightmare on Buccleuch Street. Staycation post No 4. I will be out of the virtual office until 30 August. I haven’t been into the university since the pandemic began but recently I dreamt that I was in the university library, in the section where magazines and journals are kept. In this dream, I was sittingContinue reading Nightmare on Buccleuch Street.

Why am I so concerned about Onsager’s so-called conjecture? Staycation post No 3. I will be out of the virtual office until 30 August. In recent years, Onsager’s (1949) paper on turbulence has been rediscovered and its eccentricities promoted enthusiastically, despite the fact that they are at odds with much well-established research in turbulence, beginningContinue reading Why am I so concerned about Onsager’s so-called conjecture?

That’s the giddy limit! Staycation post No 2. I will be out of the virtual office until 30 August. The expression above was still in use when I was young, and vestiges of its use linger on even today. It referred, often jocularly, to any behaviour which was deemed unacceptable. Why giddy? I’m afraid thatContinue reading That’s the giddy limit!

When is a conjecture not a conjecture? Staycation post No 1. I will be out of the virtual office until 30 August. That sounds like the sort of riddle I used to hear in childhood. For instance, when is a door not a door? The answer was: when it’s ajar! [1] Well, at least weContinue reading When is a conjecture not a conjecture?

How do we identify the presence of turbulence? In 1971, when I began as a lecturer in Engineering Science at Edinburgh, my degree in physics provided me with no basis for teaching fluid dynamics. I had met the concept of the convective derivative in statistical mechanics, as part of the derivation of the Liouville equation,Continue reading How do we identify the presence of turbulence?

Are Kraichnan’s papers difficult to read? Part 2: The DIA. In 2008, or thereabouts, I took part in a small conference at the Isaac Newton Institute and gave a talk on the LET theory, its relationship to DIA, and how both theories could be understood in terms of their relationship to Quasi-normality. During my talk,Continue reading Are Kraichnan’s papers difficult to read? Part 2: The DIA.

Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance. When I was first at Edinburgh, in the early 1970s, I gave some informal talks on turbulence theory. One of my colleagues became sufficiently interested to start doing some reading on the subject. Shortly afterwards he came up to me at coffee time and said.Continue reading Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance

Hurrah for arXiv.com! In my previous blog, I referred to my paper with Michael May [1], which failed to be accepted for publication, despite my having tried several journals. I suppose that some of my choices were unrealistic (e.g Nature) and that I could have tried more. Also, I could have specified referees, which IContinue reading Hurrah for arXiv.com!

The Kolmogorov (1962) theory: a critical review Part 2 Following on to last week’s post, I would like to make a point that, so far as I know, has not previously been made in the literature of the subject. This is, that the energy spectrum is (in the sense of thermodynamics) an intensive quantity. ThereforeContinue reading The Kolmogorov (1962) theory: a critical review Part 2

The Kolmogorov (1962) theory: a critical review Part 1 As is well known, Kolmogorov interpreted Landau’s criticism as referring to the small-scale intermittency of the instantaneous dissipation rate. His response was to adopt Obukhov’s proposal to introduce a new dissipation rate which had been averaged over a sphere of radius , and which may beContinue reading The Kolmogorov (1962) theory: a critical review Part 1

The Landau criticism of K41 and problems with averages The idea that K41 had some problem with the way that averages were taken has its origins in the famous footnote on page 126 of the book by Landau and Lifshitz [1]. This footnote is notoriously difficult to understand; not least because it is meaningless unlessContinue reading The Landau criticism of K41 and problems with averages

The Kolmogorov-Obukhov Spectrum. To lay a foundation for the present piece, we will first consider the joint Kolmogorov-Obukhov picture in more detail. For completeness, we should begin by mentioning that Kolmogorov also used the Karman-Howarth equation, which is the energy balance equation connecting the second- and third-order structure functions, to derive the so-called `‘ lawContinue reading The Kolmogorov-Obukhov Spectrum.

Why do we call it ‘The Kolmogorov Spectrum’? The Kolmogorov spectrum continues to be the subject of contentious debate. Despite its great utility in applications and its overwhelming confirmation by experiments, it is still plagued by the idea that it is subject to intermittency corrections. From a fundamental view this is difficult to understand becauseContinue reading Why do we call it ‘The Kolmogorov Spectrum’?

The different roles of the Gaussian pdf in Renormalized Perturbation Theory (RPT) and Self-Consistent Field (SCF) theory. In last week’s blog, I discussed the Kraichnan and Wyld approaches to the turbulence closure problem. These field-theoretic approaches are examples of RPTs, while the pioneering theory of Edwards [1] is a self-consistent field theory. An interesting differenceContinue reading The different roles of the Gaussian pdf in Renormalized Perturbation Theory (RPT) and Self-Consistent Field (SCF) theory.

What if anything is wrong with Wyld’s (1962) turbulence formulation? When I began my PhD in 1966, I found Wyld’s paper [1] to be one of the easiest to understand. However, one feature of the formalism struck me as odd or incorrect, so I didn’t spend any more time on it. But I had foundContinue reading What if anything is wrong with Wyld’s (1962) turbulence formulation?

Is turbulence research still in its infancy? Recently I came across the article by Lumley and Yaglom which is cited below as [1]. I think it is new to me but quite possibly I will find it lurking in my filing system when at last I am able to visit my university office again. ItContinue reading Is turbulence research still in its infancy?

Culture wars: applied scientists versus natural scientists. In my early years at Edinburgh, I attended a seminar on polymer drag reduction; and, as I was walking back with a small group, we were discussing what we had just learned. In response to a comment made by one member of the group, I observed that itContinue reading Culture wars: applied scientists versus natural scientists.

‘A little learning is a dangerous thing!’ (Alexander Pope, 1688-1744) I have written about the problems posed by the different cultures to be found in the turbulence community; and in particular of the difficulties faced by some referees when confronted by Fourier methods. My interest in the matter is of course the difficulties faced byContinue reading ‘A little learning is a dangerous thing!’ (Alexander Pope, 1688-1744)

Intermittency, intermittency, intermittency! It is well known that those who are concerned with the sale of property say that the three factors determining the value of a house are: location, location, location. In fact I believe that there is a television programme with that as a title. This trope has passed into the general consciousness;Continue reading Intermittency, intermittency, intermittency!

Does the failure to use spectral methods harm one’s understanding of turbulence? Vacation post No. 3: I will be out of the virtual office until Monday 19 April. As described in the previous post, traditional methods of visualising turbulence involve vaguely specified and ill-defined eddying motions whereas Fourier methods lead to a well-defined problem inContinue reading Does the failure to use spectral methods harm one’s understanding of turbulence?

Does the use of spectral methods obscure the physics of turbulence? Vacation post No. 2: I will be out of the virtual office until Monday 19 April. Recently, someone who posted a comment on one of my early blogs about spectral methods (see the post on 20 February 2020), commented that a certain person hasContinue reading Does the use of spectral methods obscure the physics of turbulence?

Stirring forces and the turbulence response. Vacation post No. I: I will be out of the office until Monday 19 April. In my previous post, I argued that there seems to be really no justification for regarding the stirring forces that we invoke in isotropic turbulence as mysterious, at least in the context of statisticalContinue reading Stirring forces and the turbulence response.

The mysterious stirring forces In the late 1970s there was an upsurge in interest in the turbulence problem among theoretical physicists. This arose out of the application of renormalization group (RG) methods to the problem of stirred fluid motion. As this problem was restricted to a very low wavenumber cutoff, these approaches had nothing toContinue reading The mysterious stirring forces

Is the entropy of turbulence a maximum? In 1969 I published my first paper [1], jointly with my supervisor Sam Edwards, in which we maximised the turbulent entropy, defined in terms of the information content, in order to obtain a prescription for , the renormalized decay time for the energy contained in the mode withContinue reading Is the entropy of turbulence a maximum?

Analogies between critical phenomena and turbulence: 2 In the previous post, I discussed the misapplication to turbulence of concepts like the relationship between mean-field theory and Renormalization Group in critical phenomena. This week I have the concept of ‘anomalous exponents’ in my sights! This term appears to be borrowed from the concept of anomalous dimensionContinue reading Analogies between critical phenomena and turbulence: 2

Analogies between critical phenomena and turbulence: 1 In the late 1970s, application of Renormalization Group (RG) to stirred fluid motion led to an upwelling of interest among theoretical physicists in the possibility of solving the notorious turbulence problem. I remember reading a conference paper which included some discussion that was rather naïve in tone. ForContinue reading Analogies between critical phenomena and turbulence: 1

Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis. Earlier this year I received an enquiry from Alex Liberzon, who was puzzled by the fact that some people plot temporal frequency spectra with a power law, but he was unable to reconcile the dimensions. This immediately took me back to the 1970s when IContinue reading Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis.

The concept of universality classes in critical phenomena. The universality of the small scales, which is predicted by the Richardson-Kolmogorov picture, is not always observed in practice; and in the previous post I conjectured that departures from this might be accounted for by differences in the spatial symmetry of the large scale flow. To takeContinue reading The concept of universality classes in critical phenomena.

Macroscopic symmetry and microscopic universality. The concepts of macroscopic and microscopic are often borrowed, in an unacknowledged way, from physics, in order to think about the fundamentals of turbulence. By that, I mean that there is usually no explicit acknowledgement, nor indeed apparent realization, that the ratio of large scales to small scales is manyContinue reading Macroscopic symmetry and microscopic universality.

Can statistical theory help with turbulence modelling? When reading the book by Sagaut and Cambon some years ago, I was struck by their balance between fundamentals and applications [1]. This started me thinking, and it appeared to me that I had become ever more concentrated on fundamentals in recent years. In other words, I seemedContinue reading Can statistical theory help with turbulence modelling?

The last post … of the first year! A year ago, when I began this blog, few of us can have had any idea of what the year had in store from the coronavirus, now known to us as covid-19. Over the years, I have sometimes reflected on the very fortunate lives of my generation.Continue reading The last post … of the first year!

How important are the higher-order moments of the velocity field? Up until about 1970, fundamental work on turbulence was dominated by the study of the energy spectrum, and most work was carried out in wavenumber space. In 1963 Uberoi measured the time-derivative of the energy spectrum and also the dissipation spectrum, in grid turbulence; andContinue reading How important are the higher-order moments of the velocity field?

How big is infinity? In physics it is usual to derive theories of macroscopic systems by taking an infinite limit. This could be the continuum limit or the thermodynamic limit. Or, in the theory of critical phenomena, the signal of a nontrivial fixed point is that the correlation length becomes infinite. Of course, what weContinue reading How big is infinity?

My life in wavenumber space In September 1966, when I began work on my PhD, I almost immediately began to dwell in wavenumber space. After a brief nod to the real-space equations, I had to learn about Fourier transformation of the velocity field, with the wave-vector replacing the position vector , and the Navier-Stokes equationsContinue reading My life in wavenumber space

How many angels can dance on the point of a pin? When I was young this was often quoted as an example of the foolishness of the medieval schoolmen and the nonsensical nature of their discussions. I happily classed those who debated it along with those who, not only believed that the sun was pulledContinue reading How many angels can dance on the point of a pin?